Communications in Mathematical Sciences

Volume 16 (2018)

Number 8

An equation-free approach for second order multiscale hyperbolic problems in non-divergence form

Pages: 2317 – 2343



Doghonay Arjmand (Computational Mathematics and Numerical Analysis (ANMC), Section de Mathématiques, École Polytechniques Fédérale de Lausanne, Switzerland)

Gunilla Kreiss (Department of Information Technology, Uppsala University, Uppsala, Sweden)


The present study concerns the numerical homogenization of second order hyperbolic equations in non-divergence form, where the model problem includes a rapidly oscillating coefficient function. These small scales influence the large scale behavior, hence their effects should be accurately modelled in a numerical simulation. A direct numerical simulation is prohibitively expensive since a minimum of two points per wavelength are needed to resolve the small scales. A multiscale method, under the equation-free methodology, is proposed to approximate the coarse scale behaviour of the exact solution at a cost independent of the small scales in the problem. We prove convergence rates for the upscaled quantities in one as well as in multi-dimensional periodic settings. Moreover, numerical results in one and two dimensions are provided to support the theory.


multiscale methods, homogenization, wave propagation

2010 Mathematics Subject Classification

35B27, 65L12, 74Q10

Received 30 August 2017

Accepted 5 October 2018

Published 18 April 2019